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Math 545 Homework # 5: due Mar 22, 2017 1. For an n × n Fourier matrix F : (a) calculate F F , and use this to find F −1 ; (b) find a permutation of the columns of F that produces F , so that F P = F ; (c) use these to find F 2 and F 4 . 2. Let w be a primitive sixth root of unity, w6 = 1. Then w2 is a cube root. Factorize the Fourier matrix F6 into the block form I D F3 0 F6 = P, I −D 0 F3 where P is a permutation matrix; express D and F3 using powers of w. 3. Compute y = F8 c by the three steps of the Fast Fourier Transform, for each of c = (1, 0, 1, 0, 1, 0, 1, 0) and c = (0, 1, 0, 1, 0, 1, 0, 1). 4. For any n, find the determinant of the permutation matrix Pn with 1s on the reverse diagonal: 0 0 0 1 0 0 1 0 P4 = 0 1 0 0 . 1 0 0 0 5. Calculate the determinant of the 4 × 4 Vandermonde 1 a a 2 a3 1 b b2 b3 V4 = 1 c c2 c3 1 d d2 d3 matrix, .